Geodesic equivalence of metrics as a particular case of integrability of geodesic flows

We consider the recently found connection between geodesically equivalent metrics and integrable geodesic flows. If two different metrics on a manifold have the same geodesics, then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution, and vice versa. The quantum version of this result is also true: if two metrics on one manifold have the same geodesics, then the Beltrami Laplace operator Δ for each metric admits sufficiently many linear differential operators communiting with Δ. This implies that the topology of a manifold with two different metrics with the same geodesics must be sufficiently simple. We also have that the nonproportionality of the metrics at a point implies the nonproportionality of the metrics at almost all points. In memory of Mikhail Vladimirovich Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 285–293, May, 2000.     

Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems

We show that all the Antonowicz-Fordy type coupled KdV equations have the same symmetry group and similar bi-Hamiltonian structures. It turns out that their configuration space is , where is the Bott-Virasoro group of orientation preserving diffeomorphisms of the circle, and all these systems can be interpreted as equations of a geodesic flow with respect to L² metric on the semidirect product space .     

Geodesic flows on path spaces

On the path space over a compact Riemannian manifold, the global existence and the global uniqueness of the quasi-invariant geodesic flows with respect to a negative Markov connection are obtained in this paper. The results answer affirmatively a left problem of Li.     

Geodesic flows on path spaces

On the path space over a compact Riemannian manifold, the global existence and the global uniqueness of the quasi-invariant geodesic flows with respect to a negative Markov connection are obtained in this paper. The results answer affirmatively a left problem of Li.     

Darboux-Jouanolou-Ghys integrability for one-dimensional foliations on toric varieties

We use the existence of homogeneous coordinates for simplicial toric varieties to prove a result analogous to the Darboux-Jouanolou-Ghys integrability theorem for the existence of rational first integrals for one-dimensional foliations.     

Rigid geometry on projective varieties

We prove rigidity of various types of holomorphic geometric structures on smooth complex projective varieties.     

Geodesic flows on Riemannian g.o. spaces

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.     

Analyticity and flows in von Neumann algebras

Let $\text{B}$ be a von Neumann algebra, let {α_t}_tεR be an ultraweakly continuous one-parameter group of ¹-automorphisms of $\text{B}$, and let $\text{U}$ be the set of all A such that for each ? in $\text{B}$₁, the function t → ?(α_t(A)) lies in H^∞($\text{R}$. Then $\text{U}$ is an ultraweakly closed subalgebra of $\text{B}$ containing the identity which is proper and non-self-adjoint if {α_t}_tεR is not trivial. In this paper, a systematic investigation into the structure theory of $\text{U}$ is begun. Two of the more note-worthy developments are these. First of all, conditions under which $\text{U}$ is a subdiagonal algebra in $\text{B}$, in the sense of Arveson, are determined. The analysis provides a common perspective from which to view a large number of hitherto unrelated algebras. Second, the invariant subspace structure of $\text{U}$ is determined and conditions under which $\text{U}$ is a reductive subalgebra of $\text{B}$ are found. These results are then used to produce examples where $\text{U}$ is a proper, non-self-adjoint, reductive subalgebra of $\text{B}$. The examples do not answer the reductive algebra question, however, because although ultraweakly closed, the subalgebras are weakly dense in $\text{B}$.     

Complex varieties and higher integrability of Dir-minimizing Q-valued functions

We provide new elementary proofs of the following two results: every complex variety is locally the graph of a Dir-minimizing function, first proved by Almgren (Almgren’s big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics, 2000); the gradients of Dir-minimizing functions, in principle square-summable, are p-integrable for some p > 2, proved by De Lellis and Spadaro (Higher integrability and approximation of minimal currents, 2009). In the planar case, we prove that our integrability exponents are optimal.     

Secant varieties and birational geometry

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties. Received: 29 November 1999; in final form: 4 September 2000 / Published online: 23 July 2001     

Semiclassical geometry and integrability of the AdS/CFT correspondence

We discuss the semiclassical geometry and integrable systems related to the gauge—string duality. We analyze semiclassical solutions of the Bethe ansatz equations arising in the context of the AdS/CFT correspondence, comparing them to stationary phase equations for the matrix integrals. We demonstrate how the underlying geometry is related to the integrable sigma models of the dual string theory and investigate some details of this correspondence.Translated from Teoreticheskaya Matematicheskaya Fizika, Vol. 142, No. 2, pp. 265–283, February, 2005.     

On the integrability and perturbation of three-dimensional fluid flows with symmetry

Summary The purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.     

Planar quasi-homogeneous polynomial differential systems and their integrability

In this paper we study the quasi-homogeneous polynomial differential systems and provide an algorithm for obtaining all these systems with a given degree. Using this algorithm we obtain all quasi-homogeneous vector fields of degree 2 and 3.     

Singular Poisson-Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.